Problem 83 - Precalculus with Limits

Question

In Exercises 81 - 85, determine whether the statement is true or false. Justify your answer. If the second differences of a sequence are all zero, then the sequence is arithmetic.

Step-by-Step Solution

Answer

The statement is false. Although the arithmetic sequence does have a second difference of zero, not all sequences with a second difference of zero are arithmetic sequences, they can also be sequences modeled by a quadratic function, as shown in the example of the sequence of squares.

Step 1: Define Second Differences

Second differences are calculated by finding the differences between consecutive terms and then finding the differences of these differences.

Step 2: Understand Arithmetic and Non-Arithmetic Sequences

In an arithmetic sequence, the difference between consecutive terms is a constant. But not all sequences with zero second differences are arithmetic. Instead, they can be modeled by a quadratic function.

Step 3: Example of Second Differences of Arithmetic Sequence

Consider the arithmetic sequence 2, 4, 6, 8, 10. The first differences are 2 (4-2), 2 (6-4), 2 (8-6), 2 (10-8), which are all constant. The second differences, which are the differences of the first differences, are all 0 (2-2, 2-2, 2-2). This is consistent with the statement.

Step 4: Counter example of Second Differences of Non-arithmetic Sequence

Consider the quadratic sequence 1, 4, 9, 16, 25 (sequence of squares). The first differences are 3 (4-1), 5 (9-4), 7 (16-9), 9 (25-16), which are not constant. The second differences, however, are 2 (5-3), 2 (7-5), 2 (9-7), which are all constant. But this sequence is not arithmetic.

Key Concepts

Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This fixed amount is known as the 'common difference'.
For example, in the sequence 3, 6, 9, 12, the common difference is 3, since each term is 3 more than the previous term.
Arithmetic sequences are linear in nature, meaning if you plot them on a graph, they form a straight line.

• Recognizing an arithmetic sequence means checking that the difference between every pair of consecutive terms remains the same throughout.
• To find the nth term of an arithmetic sequence, simply use the formula: \( a_n = a_1 + (n-1)d \), where:
  • \( a_n \) is the nth term,
  • \( a_1 \) is the first term, and
  • \( d \) is the common difference.

Remember, in an arithmetic sequence, the first differences (differences between each pair of consecutive terms) will always be the same, and as a result, the second differences will be zero.

Exploring Non-Arithmetic Sequences

Non-arithmetic sequences do not have a constant difference between consecutive terms. These sequences can vary in numerous ways, either increasing or decreasing but not at a fixed rate.
By looking at the step-by-step solution, we learn that sequences like 1, 4, 9, 16, 25 are non-arithmetic. This sequence actually represents a set of squared numbers.
In these cases, the first differences are not constant, but the second differences might be constant.

• Second differences are the differences between consecutive first differences in a sequence.
• The existence of constant second differences often indicates a quadratic pattern in the sequence, which is distinct from the linear pattern of an arithmetic sequence.

While the constant second differences might seem to suggest arithmetic behavior, they actually point to the presence of a quadratic function driving the sequence. Hence, not all sequences with zero second differences are arithmetic.

Quadratic Functions and Their Representations

A quadratic function is one that can be expressed in the form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).
Quadratic functions are related to sequences where the second differences are constant. The shape of their graph is a parabola, which can open upward or downward depending on the sign of \( a \).
These functions capture the essence of non-linear growth or decline within a sequence.

• To determine if a sequence can be modeled by a quadratic function, calculate second differences. If they are constant, the sequence likely follows a quadratic pattern.
• An example of a quadratic sequence is the sequence 1, 4, 9, 16, 25, where each term can be derived as \( n^2 \), indicating that it follows the quadratic function form \( y = n^2 \).

Quadratic sequences share the feature of constant second differences, thus providing a useful tool for distinguishing them from arithmetic sequences. Understanding this distinction helps us see why some sequences with zero second differences do not fit the arithmetic mold.